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Saturday, November 21st 2015


Isis and its Expansion, part 1

  • STATE OF EXISTENCE: Moving on fumes, but moving ...

 (Please note that “ISIS,” ”ISIL”, “Daesh” and “the Islamic State” are being used synonymously for the same entity, and I will give myself flexibility as needed.)

This series of entries is intended to provide some further information concerning the militant groups of Islam who are creating havoc around the world. In the first installment I’m going to refer to some remarks made by our president and secretary of state, but only to illustrate certain attitudes. This is not intended to be a political topic per se, though it has to start that way. My basic point should be non-partisan as far as American politics are concerned.

1. Correcting a Misperception.

I just ran across the site “Truth-O-Meter,” and it appears to do a pretty good job analyzing politicians’ utterances. I’m not sufficiently familiar with it yet to know whether it has a distorting bias, but alone their reproduction of direct quotes in context makes it helpful. Here’s an example:

When President Obama said on the very day before the Paris bombings that “ISIS was contained,” given the specific aspect he was addressing, he was correct. Maybe he was still riding high on the waves of the execution of Jihadi John and various other minor successes, but those things had nothing to do with that statement. If you look at the television screen of “Good Morning America” pictured in one of the T-O-M  pages, you will see a good illustration of how his statement was misunderstood,

President Obama Doesn’t Think ‘ISIS is Gaining Strength’

the little inset informs us. Taken in any way other than the specific context in which he made the statement in question, it would have been outrageously ignorant or deceptive. However, what the president was referring to was clearly limited to the geographical expansion of the territory ISIS is claiming within the countries of Syria and Iraq. For a while now, ISIS has not been able to stretch those boundaries, and that was what Mr. Obama was alluding to. He was quite aware of and acknowledged the growth of ISIS in territory and power in other parts of the world.

And that’s what I want to write about here, though I’m afraid I need to address some more matters beforehand.

2. Exposing a Misrepresentation.

On May 5, 2011, I closed my obituary of Osama bin Laden with the following statements (edited just a little), hazarding some guesses of what the future might bring:

Someone else will come along, maybe not immediately, to fill Osama's shoes, and the fact that we don't know yet whether they will be sandals, wingtips, or boots introduces a dangerous element of uncertainty. His death is a setback for al-Qaeda.  But only a setback.  There are too many people, too many cells, too many individuals whose brains have become pickled with Qutbite rhetoric for anything to have changed in the long run. I'm afraid that we have not heard the last of al-Qaeda and its nefarious activities. This is my opinion, and I can't imagine that too many people would seriously think otherwise.

Well, of course I was wrong. Some people have attempted to convince us otherwise. The Obama administration has attempted to use the assassination of bin Laden as a license to claim that they had overcome al-Qaeda.

Somewhat aside: One speculation concerning the pretzel of misinformation that we have come to know as Benghazi Gate is that President Obama allowed (or authorized?) the misrepresentation of the facts in that case because the reality, viz. that the miscreant behind bombing the American consulate belonged to al-Qaeda, would have distracted from this supposed victory. (BTW, Ms. Susan Rice, the former ambassador to the U.N., who spread the false cover story and then blamed the intelligence services, is now one of Mr. Obama’s chief foreign policy advisers.)  

One might have thought that, as things were developing over the subsequent few years, Obama’s people would have silenced their trumpets on this matter, but until this week that has not happened. In what I wrote at the outset of this post I agreed that the president’s statement concerning the “containment” of ISIS had been misconstrued, and that within its proper context it was correct and appropriate. No such exoneration is possible for the current Secretary of State, John Kerry, who said on the day before the bombings in Mali that “they” (the Obama administration?) had neutralized al-Qaeda. Here is the direct quote,

It took us quite a few year before we were able to eliminate Osama Bin Laden and the top leadership of Al-Qaeda and neutralize them as an effective force. We hope to do Daesh much faster than that. We think we have the ability to do that.  

The statement was preceded by a caution that dealing with these terrorist organizations takes time, and that people need to be patient. However, Secretary Kerry seems to know (I do not see how) that with regard to ISIS (Daesh) the time table will be faster. I cannot help but see his declaration as false, irresponsible, and illiterate.

Illiterate: What in the world does it mean to “do” Daesh? What kind of a Secretary of State uses such street jargon in a public statement?
False: The fact that al-Qaeda has not been neutralized as an effective force has been obvious to most of us for quite some time now and didn’t need to be confirmed by the bombings in Mali on the day after Mr. Kerry’s speech. The atrocities didn’t tell us anything new about al-Qaeda, but they did reveal that our Secretary of State appears to be out of touch with reality.
Irresponsible: Consequently, his promise that ISIS will be eliminated (if that is what “do” means) more expeditiously than al-Qaeda is not only groundless, it’s meaningless.  But it attempts to convey a false sense of reassurance. (One is reminded of his presidential campaign, in which he insisted time and again that he was a better candidate than George W. Bush because he had a "plan," but he never told us what the "plan" was.)

And, thus we come to the important question of how ISIS is continuing to grow and become a veritable rival to al-Qaeda.

I am beginning to realize that, once again, this matter is going to take more than one installment, and I am running out of steam fast. Working through the political material took more out of me than I had anticipated; it takes a lot of energy for me to write in a civil manner about Mr. John Kerry, the person and the politician.

Next time, I’m going to bring up certain events in Nigeria as a case in point of how ISIS is increasing in power and influence.

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Wednesday, November 18th 2015


The Golden Triangle and its Offspring

  • STATE OF EXISTENCE: Once again, pretty exhausted

We haven't had too many typical November days so far this year. There's been a lot of sunshine and it's stayed relatively warm so far--high 50s and low 60s for the most part. Today (Wednesday), when you look outside, it's definitely November. It's still surprisingly warm, but all the skies are gray, the leaves are gone, and there are four or more strong winds outside.

June has been struggling quite a bit after her last dentist appointment. Without going into details, it appears to me that they crammed what should have been two appointments into one, and she is still hurting pretty badly.

I have entitled my StreetJelly set for tomorrow (Thursday) night (9pm EST) "Blame it on the Bossa Nova" in recognition of some wonderful people in my audience who either live in Brazil or have had some connection to that country. Right now, I am totally resisting the urge to write about the distinctiveness of the Bossa Nova rhythm, but I'm intrigued by it, and so it will come up eventually. I'll undoubtedly mention it tomorrow night. (In the same context, I'm also refraining from writing about the Portuegese language, which has me fascinated from a linguistic point of view.) In addition, on that next StreetJelly set I will include some recognition of the people of Paris as they are grieving and hurting.

I do need to come back to the topic of ISIS very shortly and give you some additional information that you may not be aware of, though it's rather important for the entire picture. So, I may be oscillating between topics for a short while. Today I will slowly work up to the actual focus of these entries, the remarkable number phi (φ)


Perhaps you feel that I’m spending too much time and bandwidth on more or less relevant peripheral matters. But, you see, I’m trying to provide a background, as limited as it will be at that, which allows us to do more than simply gush at the remarkable presence of phi in nature. I guess one could view my effort here as trying to counter the attitude of "I don't know anything about math, but I know that I like phi." I teased you at the outset of this series with a clone of the famous Fibonacci series, and I will return to it. Actually, I haven't mentioned it yet directly, but knowledgeable readers recognized it immediately behind the thin disguise I gave it. However, right now I’m trying to disconnect phi from the Fibonacci numbers as much as is possible. The Fibonacci series is a remarkable function, and it is true that it converges to phi the farther you compute it. However, phi is not directly derived from the Fibonacci function—never has been and never will be. But, as I said, don’t worry; we’ll come back to it.

To the best of my knowledge, the first written record concerning the ratio that we now express with the number phi stems from Euclid of Alexandria (ca. 300 BC) in his best-seller The Elements, a book that stood for several millennia as the final authority in geometry. I will attempt to describe (please note—not prove!) Euclid’s discovery of phi, though taking a slightly different sequence of steps. (I’m following pretty closely on the heels of Mario Livio, The Golden Ratio, pp. 78-82.) But, as alluded to above, please make sure you realize that Euclid was not writing about a number, but about the proportions of lines in certain geometric figures expressed in words.

1. The Golden Triangle.

Let us examine some features of a triangle, which is sometimes called the “Golden Triangle.” Eventually I will tell you how we can derive such a triangle from another geometric form, but for the moment we can just assume that there exists such a figure among the many ideal geometric objects. It is a somewhat pointy triangle (“acute”) and its two longer sides are of equal length (“isosceles"). We’ll label the corners A, B, and C and call the three sides a, b, and c respectively. Let's also label the three angles alpha, beta, and gamma.

Gilded TriangleMy readers undoubtedly know that one feature of any (Euclidean) triangle is that its three angles add up to 180 degrees. The Golden Triangle’s base angles measure 72 degrees each, which leaves 36 degrees for the top angle. You see a picture of it on the right (copied from Wolfram Alpha and modified--or wantonly bedazzled--by me).

We could print out this picture, measure the sides of the triangle and, perhaps, come to some interesting conclusions. However, proofs in math or geometry based on physical measurements don’t count for much. Standards of measurement are invented by people, and you’re going to get different numbers if you use, say, centimeters rather than inches. Ideally, the relationships, such as the ratio of one measured line to another will be the same regardless of the calibration of our physical rulers. For example, 1 inch divided by 2 inches and 2½ cm divided by 5 centimeters both come out to ½, but the measurements on which this ratio is based will still depend on the accuracy of our instruments, which is always limited, and so we couldn't really be certain that the ratio we obtained is correct. By contrast, if ever in the course of carrying out some little household carpentry project, I wound up within 1/100 of an inch precision for all of my cuts, it would be a miracle. But in math the difference between 2.49, 2.50, and 2.51 can be crucial. There is no "level of tolerance."

However, we can resort to a mental ruler, whose fundamental unit is completely exact for our purposes, and on which we can all agree, regardless of what the usual units of length in our daily lives may be. We can posit that 1 mental unit equals exactly the length of b, the base of this triangle without taking account of its size when we drew it or even whether we drew it correctly. Obviously then, we should not be surprised by the fact that the length of b is 1 since we made it that way. Therefore, by assumption,

b = 1.

Having done that much, we still do not know the lengths of the two other sides, which are, of course, equal and, for the moment at least, we can only represent them with a variable, calling on the ever-prepared “x” to do its usual duty. [See note 1 at the bottom.] Thus,

a = x     and     c = x.

Thus, so far this is what we know about our triangle.


Golden Triangle

Length of b = 1
Length of a = x
Length of c = x

Angle α = 72°
Angle β = 72°
Angle γ = 36°

2. The First Proportion: a to b

Let us now figure out the proportion of one of the sides (we’ll pick a) to the base, b. Given the convenience of our mental ruler, the proportion ab  is equal to x1.

ab  = x1

Normally we would remove the useless 1 from this expression, since there is no difference between x1and x, but let’s allow it to stand in the redundant form, x1 for the moment because it will illustrate the ensuing point a little more clearly.

3. Opening Up the Triangle to Create Line g. 

Now we will open up our triangle.

We shall disconnect line a at point C and, using point A as a hinge, swing it downward until it has become a continuation of the horizontal base, b.


We’ll call its new point of origin on the left D. We’ll also disconnect line b at point B so that we are now only concerned with a single line instead of a triangle.

Golden Ratio

This line consists of two segments, line a (extending from point D to A) plus line b (between points A and B). We can also think of the new line as an entirety running from points D to B, and, rather than erasing the segment divider in the union of a and b, I will just draw this new line afresh and call it g.

4. The Second Proportion: g to a 

Now, we’ve already brought up proportion of a to b, which we expressed as  x1.

Since line g is the sume of lines a (value: x) and b (value: 1), the length of g is x + 1.

g = x + 1

Since the variable we assigned to line a is x, the proportion of line g to line a is  x + 1x

Larger ratio

5. The Golden Ratio

Now, it turns out that in the Golden Triangle these two proportions (a to b and g to a) come out as equal, and this is the "Golden Ratio." Euclid called it the proportion of the mean to the extreme. Expressed in words, it says that

The proportion of the larger segment (a) to the smaller one (b) 

is equal to

the proportion of the entire line (g) to the larger segment (a).

As soon as we are stating it in its algebraic form, we are doing something that Euclid probably never dreamed of, given the clumsy letter-based system of numbers the ancient Greeks had to work with.

golden ratio

6. Working Towards Finding a Value for φ 

Substituting the one variable (x) and the one value we have (1) we have for the names of the lines (a, b, g), we get:

Golden Ratio

Now that it has served its purpose of displaying its position in setting up the ratio, we can drop that silly 1 on the left side, and we get

golden ratio formula

Let’s eliminate the fraction on the right side by multiplying both sides by x. Then we have:

x2 = x + 1.

We can subtract (x + 1) from both sides [or add –(x + 1), if you want to look at it that way], and we have arrive at:  

x2  – x – 1 = 0.

This kind of equation is called a "quadratic equation," and there are several related ways of solving it. The most famous one is a formula, which is not all that hard to manipulate, but which may strike you as quite intimidating before you've made friends with it. Someone once gave me a mnemonic for it, but I've forgotten it. If anyone know one, please let me know. I wonder if I need a mnemonic for the mnemonic.

Deliberation on the Sideline: Should I bring up the formula here and now? I'm afraid that, regardless of what I write here,  I'll probably never run out of visitors to this blog because there will always be the spammers who like to peddle their wares via the comments sections of my blog, for which I pay to keep it ad-free. A highly prominent subgroup among these stowaways are some gentlemen from Malaysia and Indonesia attempting to lure one into purchasing tours of the world, the luggage you need to travel, and ways of enhancing the companionship you might encounter on such a trip. There are also the ladies from Quebec, fortune tellers one and all, who are eager to use their gifts of clairvoyance on your behalf without asking for anything in return. However, my discussions are not really geared to these friends of free enterprise. I don't know how many people are actually reading any of my streams of conscious with a more serious intent, though I can't bring myself to believe that they could all be spammers. Still, I fear that, based on how it looks at first, if I bring up the formula and plug in its values here and now, it could cut back on my already fragile readership. People who are at home with math probably know more than I do, and don't need to read this. Those who do not could just get scared away by it. Then my Malaysian, Indonesian, and French-Canadian interlopers might just be my only constituency. It's probably best if I leave the formula and its application to a note. Please see note 2 below. 

Skipping over the calculations then, I must tell you now that all quadratic equations have two solutions or "roots," one positive and one negative. Given all of the previous discussion, I'm sure you won't be surprised that the result on computing the ratio is going to result in an irrational number, viz. one with a never-ending string of digits after the decimal points that do not present a predictable pattern. If you use the Wolfram Alpha program, it clarifies for you that the result it gives is only a "decimal approximation"; and while it only carries it to 52 places, but offers to find a longer string should you happen to need it. Livio gives us the first 2,000 digits. His list starts in the middle of page 81 and ends in the middle of page 82. Still, no matter how overwhelmed we may feel looking at this string, an approximation is what it is.

Well, I haven't exactly hidden the fact that the positive number of the ratio derived from the Golden triangle is phi. Here is Wolfram Alpha's "approximation" for the positive root of the equation.


I stated a moment ago, that a quadratic equation comes with both a positive and negative root. Now when we look at the negative root, things start to get weirdly fascinating. The number starts with a 0 rather than a 1 before the decimal point, and it is a negative fraction, but the decimal digits are the same as the ones for the positive solution. 


While we’re at it, let me also mention that there is a number called the "Golden Ratio conjugate," which does not really present us with any surprises. It is defined as φ - 1. This is anything but startling. After all, any number from which we subtract 1 will be 1 less in magnitude. 1.5 - 1 = 0.5. Not worth a letter to the editor or a tweet to the world. 

So, we have

1.61803... - 1,

and its value is quite obviously everything that comes after the numeral 1 and the decimal point. Just for fun (and for visual enhancement, I'll write out the Wolfram approximation: 


Please don't start to yawn just because I mentioned one thing that happened be clear and obvious. Yes, there's no cause for astonishment in the fact that a number minus 1 will be one less than it was before. But the very innocuous nature of this fact highlights the surprise in what follows. 

A bit of nomenclature. Any number used as the denominator of a fraction in which the numerator is 1 is called the "reciprocal" of the number. The reciprocal of 7 is 1/7 , which can also be written as 7-1. We can calculate reciprocals, of course, and for 1/the outcome in decimal notation is 0.14285714..., with that exact sequence repeating indefinitely (and thus making it a rational number). 

Now let's look at the reciprocal of phi, 1/φ, and calculate its value. Here it is, once again availing ourselves of Wolfram's "decimal approximation."


Yes, this is the same number as phi's conjugate. Phi minus 1 yields the same number as 1 over phi. No other number behaves that way. Thus:

φ - 1 = 1/φ

 Let's do one more thing, namely to square phi: The value of φ2  is:  


As you can readily see, it's precisely the same as phi, only larger by exactly 1.

So, these things are true of phi and no other number: 

The negative root of the equation that produces it has the identical decimal points as phi. (-0.618034)
The conjugate of phi is exactly 1 less than phi.
The reciprocal of phi is exactly 1 less than phi, and thus equal to its conjugate.
The square of phi is exactly 1 more than phi.  

Other numbers have their own peculiarities, which may be just as startling, but the traits that you see here belong to phi alone.

Are you beginning to see now why I’m talking about beauty within the numbers themselves? Why numbers sometimes appear to have personalities, at least for those of us who have a little bit of inclination towards fantasy? That a number can be distinct from all others in more than a trivial sense, and that some are more eccentric (or gifted?) than others? That you don’t have to count generations of rabbits or petals of a rose blossom to find fascinating and startling facts about phi (though it may be fun, and we'll get to that)? Other numbers have their own peculiarities, which are just as startling, but the traits that you see here belong to phi alone. The beauty is already there in the number. The rest, which don't want to minimize but just ignore right now, is over and above the properties of the number alone. 

Sorry, we're still not ready to talk about Fibonacci and his series of numbers; we need to stay with its geometrical home for a little longer. In this entry we have stipulated the Golden Triangle, knowing in advance how the ratios and numbers would fall out. The question we need to address is whether this geometric figure is anything other than something concocted by an ancient mathematician for the entertainment of his guests on long November evenings. Can we find the Golden Triangle somewhere where it is right in place, playing a significant role in geometry?

Let's go to the Pentagon to find out!



Note 1: Since we know that both a and c are of a different length than b, we could also use another approach and let x represent the difference between b and either a or c. Then we could say that a and c are of length 1+x, where x could be a negative number or zero, just in case that the "Golden Triangle should turn out to be either obtuse or equilateral. This is not a serious consideration, however, since we already know from the values for the angles that the triangle must be isosceles and acute. So, there's no need to add that complication.

Note 2: This is a quadratic equation, and if you’ve had a bit of math, you may have spent some time puzzling them out. They have the general form

ax2 + bx  + c = 0

The two solutions (or “roots”) can be found with the formula:

Quadratic Formula from SOS Math

(The above image is hosted SOS Math, a very straight-forward and careful website.)

Note the “±" in the numerator. It is this dual operator that helps us find the two solutions, one positive and one negative.

a is the coefficient for x2 ;if there is none, a 1 is implied;
b is the coefficient for x; again, if there is none, a 1 is implied.
c is the modular constant; if there is none, then the value for c is 0.

In our specific case,

x2  – x – 1 = 0.

a=1, b=-1, and c=-1.

Once you go through the two stages of computation for each root (or let Wolfram Alpha do them for you), you arrive at the values for phi, as mentioned above in the text.


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Saturday, November 14th 2015


Militant Islam, part 3

  • STATE OF EXISTENCE: still tired

And now to ISIS. As announced I will not give a history of this organization (except in the sketchiest terms), nor go into all of their present machinations in detail. The Wikipedia site is flush with details and will definitely send a chill down your spine. My goal here is the rather modest one of putting ISIS on the map of Islamic groups.

One very basic truth needs to be reiterated: Islam is never complete as a religion unless it encompasses a state, viz. a political entity. (For al-Qaeda, such a state would be a pure theocracy, arrived at by ungodly measures.) Thus, the fact that ISIS has the establishment of a state as its goal should not be surprising. Still, when my father and I were talking about Islam today on our weekly Skype, we agreed that neither the media nor the vast majority of politicians has yet to catch on to the fact that Islam cannot function as a private “faith,” but must establish the community (al-ummah) as a state. Islam and Christianity are very different in that respect—or should be, and I don’t want to go further along that line for now. See my article “God in the Early Twenty-first Century.”  http://wincorduan.net/God and Ayodhya.pdf

ISIS is not Wahhabi, though there are many similarities. It promotes a supposedly “pure" kind of Islam. It identifies itself as Salafi and stresses the importance of tawhid, the oneness of God and the worship of him alone. Any practice that could be deemed to be shirk (idolatry) must be eliminated. But none of these matters require any link to Wahhabism other than a conceptual one. By engaging in aggressive warfare in order to expand its boundaries and killing Muslims as well as non-Muslims for merely political reasons, it goes far beyond Wahhabi ideology. And, for that matter, so does the genocide of non-Muslims apart from any manufactured war, which includes women and children, utilizing some of the worst tortures ever invented by monsters disguised as humans. Christians and Yezidi are easy targets for them; consequently there is now a growing community of Yezidi refuges in Germany. Please see my post about the Yezidi. http://wincorduan.bravejournal.com/entry/144001

Nor is ISIS Qutbite. The movement did arise out of al-Qaeda with its underlying Qutbite­­ ideology, and it definitely shares Qutbite radicalism in designating all other Muslims to be in a state of jahiliyyah and, thus, making them legitimate targets of aggressive war. But al-Qaeda and ISIS could not coexist for long since ISIS is all about setting up a new government, while Qutbism eschews them all. In the matter of persecuting Muslims, ISIS has emphasized fighting against what they are calling “Shi'ite oppression." This notion would be laughable if it were not accompanied by such inhumane violence and brutality. It is true that since the American involvement in Iraqi politics the Shi’ites have had a greater amount of authority in Iraq than in most of Iraq’s history. However, Shi’ites do represent a majority of Iraq’s population, and, as I stated above, they have lived almost perpetually under Sunni governments. ISIS considers itself as neither Sunni nor Shi’ite, just as was the case for the Kharijites in the seventh century, but their main victims within Islam are the Shi’ites. Outside of Islam, it appears that anyone else is fair game for persecution.

Most significantly, in contrast to both Wahhabism and Qutbism, ISIS has resurrected the position of caliph. Thus, at this moment, they occupy a unique slot. Once again consulting history, after the initial skirmishes, the Umayyad dynasty held the caliphate until it was replaced by the Abbasids in 750. The only territory to which the Umayyads held on was Iberia, and they continued to designate their leaders as caliphs. Thus there was the truly powerful Caliph of Baghdad and the vestigial Caliph of Cordoba. The Abbasids eventually lost power, and in the eleventh century, the strongest caliphate was held by the Shi’ite Fatimid dynasty in Egypt. Thus, there were now three caliphs, the Caliphs of Baghdad and Cordoba—both of them dysfunctional—and the Caliph of Cairo. After the Ottoman Empire had consolidated the Muslim world under its rule, the Sultan also bore the title of “Caliph." The revolution by the “young Turks" in 1922 (that’s where the term originated) ended that practice, and, after several failed attempts to sustain the position in some way both inside and outside of Turkey, since 1924 there has no longer been a caliph (ignoring ephemeral extravaganzas exhibited in third-world countries).

ISIS has now attempted to revive the caliphate on a global scale. Its present leader goes by several names and titles, and I will only touch on a few highlights. Born as Ibrahim Awwad Ibrahim Ali al-Badri al-Samarrai, he is the Emir (“prince" or “ruler") of ISIS. His popular name has been Abu Bakr al-Baghdadi, and you will immediately recognize the importance of “Abu Bakr" in that construction. Lately he has added “al Qurashi" to his title, implying descent from the prophet, who, you remember, was of the Quraish tribe. The shortest version with which one can refer to him is “Caliph Ibrahim."

Note: Much of the above is taken from the earlier posts with some modifications. I have not changed the passages below (except maybe for some spelling or punctuation mistakes) because it seems to me that matters have not changed all that much. ISIS has grown over the last year. Has there been progress in putting a halt to it, other than some trophy killings among their leadership?

ISIS displays all of the worst traits associated with the stereotypes of Islam, and in a world where it seems to be impolite to call an evil person "evil," surely this organization and its leader deserve that appellation. The U.S. has taken the lead in attempting to put an end to the travesties of ISIS. I’m not fond of the idea of the U.S. being the policeman of the world, but such a horror cannot be allowed to go on unimpeded. Some European countries have now joined the American effort, and the new Iraqi government is attempting to put an end to ISIS. Perhaps together they will succeed. Also, other Muslim countries have denounced ISIS.

And that last remark once again brings me to a rhetorical question that I have repeated several times. Why are the so-called moderate Muslim nations not playing a more active part in the war against ISIS? For Muslims to sit on their hands while they make frowny faces and verbally dissociate themselves from ISIS is simply not enough. If Islamic countries want to be taken seriously by the outside world, they need to earn that respect by neutralizing those groups that clearly violate the standards of Islam.

Perhaps Western-style democracy is not yet appropriate for some countries where tribalism is too deeply ingrained; simply imposing it seems to backfire in many cases, though I wish it weren’t so. But even an absolute ruler can and should abide by the Qur’an if he is a true Muslim. “There should be no compulsion in religion" (2:256). “People of the Book" (e.g., Christians and Jews) must pay the unbeliever’s tax (jizya) and occupy a lower standing in society than Muslims, but they should be allowed to live and worship God in their own way (9:29). (And, please, there is no way in which one can plausibly rationalize, let alone justify, Caliph Ibrahim’s and ISIS’s actions as a consequence of the Crusades. Unfortunately, if you’ve had conversations with Muslim apologists, you might not be surprised if someone at this very moment is attempting to do just that. )

Muslim leaders, if you want us to see any credibility in your claims concerning Islam, please join actively in the effort to put ISIS out of business!

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Saturday, November 14th 2015


Militant Islam, part 2

  • STATE OF EXISTENCE: just as tired

Continued from the previous post.

The Kharijites

Taking Aïsha out of the picture did not uncomplicate matters very much. The big question between the followers of Ali and Muawiyah was, "Should the caliphate belong 1) to someone who was highly regarded in a particular tribe or b) to a descendant of the prophet?" A third group, called the Kharijites (“Dissenters") emerged with the message that neither criterion was true to Islam. They observed that it had hardly been a bare thirty years after Muhammad’s death, and already the people had lost their way, treating the caliphate as though it were a kind of monarchy. They insisted that the truest and purest of all Muslims should be the one chosen to be caliph, even if he had been nothing more than a slave boy. Descent or social standing should have nothing to do with the selection. The fact that such inappropriate criteria were being used to designate the caliph, led the Kharijites to conclude that many supposed Muslims had become apostate. They had already fallen back into the ignorance and darkness of the time before Muhammad (the jahiliyyah). If so, these lapsed Muslims were considered to be worse than unbelievers and potentially subject to execution. K

The Kharijites acted on their beliefs. When Ali started to negotiate with Muawiya rather than fight him, a group of Kharijites assassinated him. They did not think of themselves as either Sunni or Shi’ite (which were still in their embryonic states).

While we are touching on the Shi’ites, it is important to realize for what follows that Shi’ite Islam has constituted the majority population of both Iran and Iraq once they became Islamic. However, the area that we now call Iraq has been governed by Sunnis for almost all of that time: the Umayyad dynasty followed by the Abbasids, the Selkuk Turks, the Ottoman Turks, the Hashemite monarchy (imposed by Great Britain), and the B’ath party of Saddam Hussein.­­­

As a distinct group the Kharijites did not last very long, but similar movements, which we can call “neo-Kharijite” have popped up again and again during the history of Islam. Please read my website, Groups of Islam, or Neighboring Faiths for more of what transpired as I will try to limit myself to only the most relevant parts here. The point is that these neo-Kharijite groups would not have arisen if Islam had been practiced in the pure, unadulterated form that they expected. Some developments within Islam went into the direction of mysticism (i.e. Sufism), as well as adaptations to folk religion and superstitions. 


One response to these alleged deviations came from Muhammad ibn Abd al Wahhab (1703-1792), who initiated a significant reform movement in what is now known as Saudi Arabia. When most of the Arabian Peninsula came under Saudi rule in the early twentieth century, “Wahhabism" became mandatory, and anyone who did not comply with its teachings was liable to be executed. It is a rather rigid form of Islam, forbidding anything that could be interpreted as idolatry (shirk), such as the veneration of Muslim saints or their gravesites, not even making an exception for the burial site of Muhammad in Medina. A central concept in Wahhabism is tawhid, which basically means “oneness” and refers to the one God (Allah) alone as being worthy of worship.

The Wahhabi movement came about in order to purify Islam. Countries that adopted it have not been hospitable to other religions or versions of Islam, but on the whole, their initiatives have been internal. In addition to Saudi Arabia, the two other countries in which it has made its home are the United Arab Emirates and Afghanistan, where the Taliban forced Wahhabi interpretations of the Qur’an on the people for a time (and would like to do so again). But there is no caliphate in Wahhabism. The kings of Saudi Arabia claim to have a divine mandate to rule, but they do not refer to themselves as caliphs.

One of the crucial tenets of Wahhabism is that no Muslim should follow human teachers in their beliefs and practices. Consequently, Wahhabis do not care for that label and prefer to be called Salafis, which means that they follow the pattern set by Islam under the first three caliphs.  We will meet the term Salafi again. Whether they like labels or not, Wahhabism is clearly a part of Sunni Islam, and, if I may give you some terms without explaining them here, they are Ash’arite in outlook and Hanbalite in their shari’a. 

Qutbism and al-Qaeda

Even though Osama bin Laden (1957-2011), the eventual leader of al-Qaeda, grew up in Saudi Arabia under Wahhabi teaching, he eventually turned into a different, far more radical direction, inspired by the writings of Seyyid Qutb (1906-1966). Qutbism and its philosophical allies deny the legitimacy of any human government (including a caliphate) and look for a totally Islamic world, which will be governed by Shari’a alone. According to them, not only non-Islamic countries, but also all Muslim countries that are governed by human beings (and they all are, without exception) are in the state of jahiliyyah (darkness and ignorance). The Saudi government is not only included in that judgment, but was singled out by Osama over and over again as a case in point of such apostasy.

I’m sorry for playing the same record so many times over the last few years, but I must continue to urge you to read Milestones by Seyyid Qutb. I’m certainly happy if you just take my word for the summary, but if there is any doubt in your mind concerning the utter destructiveness of some Islamic ideologies, this book should resolve it for you.

The Qur’an teaches that Islam should not be propagated by the sword (Sura 2:256). People should be able to come to a free decision of the truth. Consequently, many Muslim apologists today go to extraordinary lengths to explain away the various aggressive wars fought by Muslims ever since its birth. For our purposes right now, this issue is irrelevant because al-Qaeda and other followers of Seyyid Qutb not only acknowledge that there have been Muslim wars of aggression, but advocate that they must be resumed.

Qutbites agree that one should only become a Muslim by free choice when one recognizes the truth, but declare that people today do not have true freedom to make such a choice since they are “enslaved" to human governments. Thus, the primary item of their present agenda is to abolish governments, both those of non-Muslims and of the multitude of pseudo-Muslims who live within the recrudescence of jahiliyyah. In order to make a truly free choice concerning Islam, people need to live in a truly Islamic environment, and that is one in which only Shari’a is needed without people to enforce it. In short, it takes global violence in order to arrive at a state where only one ideology (Islam) rules, and then, finally, people will be free to choose whether they want to be Muslims or not.

Let me repeat a point that I’ve made several times in various places in order to illustrate this idea. It has been stated multiple times that al-Qaeda’s attack on the Twin Towers on 9/11/2000 was irrational because it not only killed non-Muslims, but Muslims as well. This particular inconsistency vanishes under the Qutbite paradigm because any supposed Muslim working in the Twin Towers would surely be in the state of jahiliyyah, and, thus, be subject to destruction just as much as any other infidel. Anyone who is not a Muslim in accord with the principles of Qutb and al-Qaeda is a hypocrite, and the Qur’an destines hypocrites to a worse fate than unbelievers. Sura 4:145: “The Hypocrites will be in the lowest depths of the Fire: no helper will you find for them; -“

Next installment: ISIS

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Saturday, November 14th 2015


Militant Islam, part 1


I’m posting this entry, which is a patchwork quilt of some earlier ones, in response to the bombings carried out by ISIS members in Paris on 11/13/2015. You cannot fight the enemy that you do not know. There are three parts, basically ready for upload, so they should all be up within an hour or so, after which I may revise a sentence or two and add a few pictures. 

Current events have created the need to interrupt my series on phi, and I suspect, if it were not for the reason why I’m doing so, there could just be a few sighs of relief from my loyal readers. Disloyal ones, if I had any, would not even have tried to get through what I have posted along that vein. But I will get back to math shortly; in fact, I have an entry all written out, but this topic should take priority.

Yesterday (Friday) afternoon I once again visited the class on world religions at Taylor University, which is taught by Dr. Kevin Diller. As I did last year, I gave an overview of radical Islam, beginning with the clashes about who would become Muhammad’s successor and ending up with the Taliban, Al-Qaeda, and ISIS. I also once again tried to include the point of view of Aïsha, one of Muhammad’s widows, who exercised a great amount of influence on the events following Muhammad’s death and the formation of the Hadith, the narrative supplement to the Qur’an. Furthermore, it is impossible to understand Aïsha’s life and actions and gloss over the violence and hate that characterized already the very first generation of Islam’s existence. How sad that, just a few hours later, as we turned on the news, we got an audio-visual lesson on militant Islam and the terrorism it engenders!

In this entry I’m combining much of what I posted last year as a background for being aware of ISIS with the material of the Aïsha lecture. ISIS was not yet in existence when I wrote the second edition of  Neighboring Faiths, and last year’s entry on Aïsha, which has more details than the book, is no longer accessible due to an apparent incompatibility of my code with Bravenet’s preferences. The current edition of Neighboring Faiths has two chapters on Islam, the second one being devoted entirely to the radical movements. So, if you would like further information on what I’m writing about here, you’ll find a lot of it—with painstaking documentation—in chapter 5. Also, please visit my lengthy website “Groups of Islam” for more background and some details that I cannot go into now. 

Please keep in mind that most complex scripture-based religions expect to be the only religion at the end of time; therefore, global ambitions per se should not be considered unusual or the mark of an invidious religion. However, there are important differences in the means of getting there. Spreading a faith by works of mercy, verbal proselytization, or relying on the direct action of God is not the same as conquering the planet for your religion by physical warfare. If you’re familiar with my previous writings on the subject, you already know that, various protestations by Muslim apologists notwithstanding, some Islamic groups subscribe to the latter method—and do so openly and without expressions of regret, let alone apology. They include both al-Qaeda and ISIS.

The biggest division within Islam is, of course, the split between Sunna and Shi’a, and it began right after Muhammad’s death. Who among those who had been his Companions (an honorific title as well as a descriptive noun) would lead the new community (al-ummah) established by the prophet? Muhammad’s son-in-law, Ali ibn Talib, not only seemed to have the inside edge, but also claimed to have received Muhammad’s designation and his spiritual powers. Nevertheless, the consensus (Sunna) went with Abu Bakr, Muhammad’s close friends and the father of Aïsha, his all-too-young and spirited wife. The rulers of the Islamic world became known as the “caliphs." Thus we have the majority, the Sunna, represented by Abu Bakr, and the “dividing party," the Shi’a, associated with Ali. The first three caliphs, Abu Bakr, Umar, and Uthman were acclaimed by the Sunna. Ali finally became the fourth caliph, and theoretically the breach might have been healed, but by that time the divisions within the Sunna and between Sunna and Shi’a were already unalterably headed towards mutual bloodshed.

Aïsha was not just sitting on the sidelines watching the events unfold, but took an active part in them. I think it’s safe to assume that among the majority of Westerners who are aware of her at all, the extent of their knowledge ends with the fact that Aïsha was betrothed to Muhammad at a very young age, and the marriage was consummated when she was still far too young for such things (six and nine years old respectively are the most commonly cited ages). Child marriage and consummation prior to puberty were a dark practice in a dark culture. Still, to understand the early history of Islam, and thereby Islam itself, after we have exercised our moral judgment, we need to look past our repugnance and learn from the facts as they were reported.

Aïsha was Muhammad’s favorite wife for the last nine years of his life, and she supported him publicly throughout that time. However, she had a head of her own right from the start and was not above criticizing the prophet himself in private, e.g., for his claims to have had revelations that allowed him to slip out of inconvenient situations. When Muhammad was on his death bed, the other wives conceded that he could stay in her quarters because they all knew that Muhammad and Aïsha had a special relationship, something that had been missing from his life during the most turbulent years of his career, which occurred (not without connection) after the death of his first wife, Khadija, his only wife as long as she lived. (There is actually no certainty on how many wives Muhammad had later on; 10 or 12 are popular options.)

After Muhammad’s death, Aïsha was very concerned that Islam should stay true to the prophet’s teachings. Because of her close relationship to him, she frequently served as consultant on the manner in which Muhammad himself had practiced what he had taught. Consequently, Aïsha became a public person, who also got involved in the ensuing political matters, including the issue mentioned above of who should be caliph.

The honor of being the first caliph went to Aïsha’s father, Abu Bakr, so things went for her as she had hoped they would. She also approved of the second caliph, Umar (another father-in-law to Muhammad), but the political aspect of the Islamic world was heading towards great turmoil. Umar was assassinated, and the next caliph, Uthman, perhaps best known for collecting the authoritative edition of the Qur’an, was not as capable a leader as his two predecessors.

An important factor that came into play was the caliph’s clan (subtribe) affiliation. Uthman came from a strong clan within the Quraish tribe, the Umayyads, while Muhammad hailed from its Hashemite segment. The Umayyads had been Muhammad’s strongest opponents and the very people who had caused the prophet to flee from Mecca to Medina (the hijra). After Muhammad had conquered Mecca, they were the last among the clans to convert to Islam.  Aïsha was not alone in thinking that a member of the Umayyad clan should not be caliph. Furthermore, it appears that Uthman’s interpersonal and leadership skills were not adequate for the job.

Aïsha thoroughly disliked Uthman. However, it is one thing to oppose the ruling caliph personally and politically, it is quite another to humiliate him and kill him and his family. The new governor of Egypt, Abdullah ibn Saad, did exactly that with the aid of a thousand or so rebels, perhaps motivated by his own ambitions toward the caliphate. Aïsha was outraged. She entertained a strong hope that the next caliph would take stern measures to avenge Uthman’s death.

The next caliph turned out to be none other than Ali ibn Talid, which made the matter all the more awkward from Aïsha’s perspective. She did not care all that much for Ali either. For one thing, he had been seeking the caliphate in competition with her father (Abu Bakr); for another, his spine was just weak enough that he could be manipulated into changing his mind on crucial issues, sometimes as though it were on a whim at the very last moment before some action should have been taken. Ali did not prosecute those who had abused Uthman, perhaps because it was not possible to identify the specific people involved; they had found refuge in the anonymity of being soldiers in Ali’s army. For Aïsha, Ali’s inaction in this case was another sign that he was no more suited for the position of caliph than Uthman had been.

There was another complication. In this, the fourth, round of choosing the caliph, Ali still had competition. It came from a man named Muawiyah, another member of the Umayyad clan, who thought that he was entitled to the Caliphate primarily because of his tribal status. So, both Ali and Muawiyah were odious to Aïsha, but she was particularly angry at Ali for letting the mistreatment of one of Muhammad’s Companions (i.e. Uthman) go unpunished.

Aïsha, now around thirty years old, gathered an army and ransacked the town of Basra, around which Ali’s troops had set up camp. Then she went on to confront Ali in pitched battle. She personally served as commander of her troops, perching on top of a camel behind the lines so that she could have a clear view of the field and give orders as necessary. Thus, this encounter has been known as the “Battle of the Camel.” Sadly for Aïsha, Ali’s troops were stronger than hers, and she lost the battle. Ali’s men captured her, and Ali had her escorted back to Mecca. She remained there for a short while, and then moved back to her original home in Medina. Her days of political involvement were over, but she taught the message of the prophet and became one of the most important sources for the secondary writings concerning Muhammad, known as the hadith. (In case you were wondering earlier, that's how we know about some of her private conversations with her husband.)

Next installment: The First Radical Muslims and their Offspring

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Tuesday, November 10th 2015


Pythagoras, Music, and Rational Numbers


First of all, here is a diagram of the different kinds of numbers: ℕ    ℤ    ℚ    ℝ    ℂ, which I discussed in the last entry.

Types of Numbers

I left off by saying that the number that is the subject of our attention, namely φ (= 1.61803…) belongs to the category of the “irrational numbers,” which, together with counting numbers (ℕ), integers (ℤ) and the rational numbers (ℚ) make up the set of so-called real numbers (ℝ). These irrational numbers received that label because they did not fit into the way in which the Early Greek mathematicians, particularly Pythagoras, attempted to envision the numerical regularity of the universe.

Pythagoras is known, among many other things, for his discovery of the linkage between numbers and music. Some people hold it against him that he supposedly mechanized the beauty of music by imposing his mathematical system on it. However, Pythagoras did nothing of the kind. He did not impose anything; he discovered that the sounds that we consider to be pleasing follow some simple numerical patterns. The music came first; Pythagoras found the numbers that appeared to be hidden in it.

I tried on a couple of recent shows on StreetJelly to demonstrate just a tiny bit of what had Pythagoras so fascinated, but it probably didn’t have as much entertainment in that context as I had hoped. So, I made a video:

Pythagoras and his followers were rather pleased with the notion that the universe could be described with ratios of whole numbers, the “rational” numbers. Now, somewhere, somehow, someone discovered that it was not so; there were numbers for which it was impossible to represent them accurately as whole-number ratios. There are many legends connected to the intrusion of irrational numbers into the beautiful harmony of the Pythagorean system. Some of them blame one of his followers named Hippasos and go so far that Pythagoras ordered his execution and had his minions swear an oath that they should never ever reveal the existence of this alleged heresy. The most we can document with reasonable assurance is that, somewhere around the time when Pythagoras flourished, someone discovered irrational numbers, and Pythagoras presumably did not much care for this idea.



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Tuesday, November 3rd 2015


Kinds of Numbers


Last time I tried to show you that the number zero, when put under a little bit of inspection, becomes a whole lot more than just a symbol for "nothing." My intention was to show that it is not just trivially true that each number has unique properties. The number that concerns us in this series of blog entries is 1.61803..., and it is one of the true stand-outs among all the numbers with which we are familiar. In popular literature it has acquired the Greek letter phi, φ, as its moniker, though academic mathematicians prefer the letter tau, τ. There are many resemblances between φ and its cousin, the even more famous pi, π.

Mathematicians recognize several classes into which various numbers fall. These groups can be visualized as concentric circles, beginning with a relatively small one and increasing in size, so that the next larger group includes all of the previous ones, but also adds many more. They are represented by hollowed-out capital letters, which are derived from their German names: ℕ, ℤ, ℚ, ℝ, and ℂ.

  is the set of all natural numbers, the ones we use to count. It does not include fractions or negative numbers. They are the whole numbers that begin with 1 and stretch to infinity. "Infinity" should not be construed as a number; the term simply designates an unending chain of numbers, and it should also be understood separately from the metaphysical concept of infinity when we apply it to God. There are two kinds of numbers among the multitude in the  circle: those that are prime and those that are not. Prime numbers are those numbers that cannot be divided without remainder by anything other than1 (which is trivial) and themselves (which is given). Any prime number divided by an integer other than 1 or itself is going to leave a fraction. For example 57 divided by 1 is 57 and divided by itself is 1, but if you try to divide it by any other whole number, you get a fraction.

61/2=30.5     61/3=20.333...     61/4=15.25     61/5=12.2    etc.

(Please note the difference between the results after division by 2, 3, and 4 in contrast to division by 3. We'll come back to that little feature shortly.)

Prime numbers have given rise to numerous interesting insights, guesses, and speculations. Among them are:   

a) the idea that there is no largest prime number (proven by Euclid around 300 BC);

b) the "Weaker" Goldbach Conjecture, which says that every odd number greater than five can be expressed as the sum of three primes, or, alternatively that every odd number greater than seven can be expressed as the sum of three odd primes (proven definitively in 2013 by Harald Andrés Helfgott, now at Göttingen);

c) the full Goldbach Conjecture, according to which every even number larger than four can be expressed as the sum of two prime numbers (so far unproven);

d) the famous Riemann Hypothesis, which is concerned with the distribution and density of prime numbers* (so far unproven).  

Serious mathematicians have been joined by seekers of fame and fortune, not to mention crackpots, to try to prove c) and d), so far probably unsuccessfully. People who are aware of their limits, as exemplified by your bloggist, may know enough to understand the problems, but are also too smart to even imagine that they could come close to a proof.

  (derived from the German word Zahl), is the next category, and it consists of all integers. It includes negative numbers, and, if there is any question of whether zero belongs into  or not, it is definitely at home within ℤ.

It might surprise you, as it did me, to know that negative numbers are a relatively recent addition to what mathematicians on the whole accept as legitimate. As late as the seventeenth century well-known mathematicians, such as Blaise Pascal dismissed them as "nonsense."

Rene Descartes approved of negative numbers only as long as they could be transformed into positive ones. Thus, if we are looking for the square roots of 1, we would say that there are two correct solutions: 1 and -1, but Descartes wanted to strike the -1, because in its unaltered state -1 is not a true number. However, he allowed the use of negative numbers in intermediate stages that will arrive at a positive number, e.g., -4 x -7 = 28.

Even in the eighteenth century, Gottfried Leibniz, the inventor of calculus as we know it, was hesitant about the use of negative numbers. Leonhard Euler (pronounced "Oiler," not "Yooler), one of the top ten mathematicians of all times, maybe even one of the top five, still felt the need to justify their use by drawing an analogy representing them as a kind of "debt."  Thus, even though  is situated logically between and the next category , historically  was accepted universally only about two thousand years after .  

 The letter is based on the word Quotient, which is also an English word, but is used in German much more commonly for the result of a division problem, similarly to the ease with which we use "sum" and "product" for addition and multiplication respectively. We call its occupants the "rational" numbers. This class contains all of the numbers that can be expressed as fractions without any unspecifiable remainders hanging around. Clearly any member of  is included in , since any whole number can be expressed as a fraction.

E.g., 17 is equal to or example, 17/1 or--perhaps a little more meaningfully--51/3.

Negative integers, who lend their distinctive flavor to belong into , though we must be careful with our signs. -17 is not equal to -51/-3 (that would come out as the positive 17). To maintain the negative sign we need to make sure that the numerator and denominator are of opposite signs (one positive, one negative). What we are adding in  is all of the neat fractions that, if divided out, eventually come to an end.  1/2, 8/25, and 256/3 are rational numbers.

So, 1/2 written in decimal form is 0.5.

8 divided by 25 yields 3.125. The remainder is 3 places long, but it comes to an end.

If we divide out the fraction 58/3, our result will never come to rest. The solution is 19.3333...., and the threes will continue interminably. But it still belongs into ℚ.  For one thing, that awkward number is still considered to be rational since it began its life as the ratio of two whole numbers, 58 and 3. For another, even though the 3s after the decimal point will go on forever, we know that they will never change. There will be 3s and nothing but 3s.

The next group is , where  stands for "real." Resist the temptation to think of  as standing for "rational." The rational numbers are already accommodated by . So, what  brings to the party is all of the irrational numbers, of which there are more than rational ones, according to Georg Cantor and modern set theory. Now, you may think of mathematics as the ultimate exercise in rationality and be somewhat befuddled as to how there could be "irrational" numbers. Well, these numbers are notirrational in the sense that they are contradictory or derived from some mystical intuition. They are "irrational" in the sense that they do not fit into the nice and tidy collection of numbers in  that can be expressed as a fraction of two integers or that, as a decimal, ever reaches a conclusion. To clarify the second point, an irrational number's decimal fractions neither come to an end point nor assume a pattern, such as the repetition of 3's we saw earlier, or even the repetition of an identical set of several numbers.  

Finally, there is , which adds the so-called imaginary numbers, or, much better, complex numbers (hence its letter designation). They do not play a role in our discussion (as far I can see now), but I'm bringing them up because it doesn't make any sense to list all of the others and leave out this last one. Above I mentioned in passing that the number 1 has two square roots, namely 1 and -1. Each of those two numbers multiplied by itself yields 1. So what would be the square root of -1? It obviously cannot be either -1 or 1 because they give us 1, not -1. I guess one could try to sneak in the notion that a square root could consist of one positive and one negative number, but doing so would fly in the face of what we mean by "square root."  So, let's just stipulate that there is such a number, and we'll call it i, for "imaginary," not "irrational," since the irrational numbers were already allocated their place in ℝ.  Philosophically, one can debate whether i is any more or less imaginary than, say, -243; some folks might say that they are equally imaginary. Others, among them your bloggist, would endow them with the same reality as all other numbers. Regardless, with i having achieved acceptance, the infinite set of real numbers (i.e. ℝ) is expanded even further. And, not-so by the way, since their first appearance in mathematics, "imaginary" numbers have also become useful in physics. They are not just applicable when they have no applicability. 

It is possible, even desirable, to combine real and imaginary numbers. But you can only do so with the same style that you would apply to a polynomial, viz. to state them as a real number plus so many i, e.g., 3+2i or 37+8i. The number "line" has become a "field" if we want to have any hope of graphing these numbers. These combinations of real and imaginary numbers gives us them the appropriate label of "complex" numbers.  As with all previous classes, includes all of the numbers enclosed by its circle, in this case the real numbers and all of their subdivisions. The numbers that do not have a need a direct reference to i can all be rewritten as "n+0i."   

Let me get back to phi now with a final comment for tonight. This number, which we shall treat in its own right eventually, does not appear in our concentric circles until we get to , the real numbers. It is neither a counting number, nor an integer in the larger sense, nor a rational number.  As we described above, that means that any expression of it as a proper fraction is only an approximation, and, for that matter, any expression of it as a decimal number is also only an approximation, though it has been calculated to thousands of places. This is one of the ways in which phi and pi resemble each other. 


Today I had my second PT, and I feel as though a truck had barreled over me (insofar as I can imagine such a thing since I've never had a truck roll over me in reality, not even a mini-pickup). We are having an unusually warm fall so far, accompanied by very little rain, so it's been nice to be outside. I wish it would last this way for many more weeks or even months, but I'm pretty sure what the realities will be.    

*The actual formulation of the Riemann hypothesis is rather opaque unless you've studied it just a bit. “All non-trivial zeroes of the zeta function have real part one-half.”  

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Sunday, November 1st 2015


Numbers: Zero and her Friends

  • STATE OF EXISTENCE: Schlepping Along

So, did you do your additions and divisions as suggested on the last post? Did you come up with that strange number 1.61803 ...?

To take things one step further, are you aware of the fact that you were implementing a slightly altered version of a recursive function that yields the so-called Fibbonaci numbers? (Don't worry if you haven't; I'm intending to explain this thing to you in this series.) Going one step further, have you been taught that the proportion of 1:1.618 is derived from this series, and that it is often called the "Golden Ratio"?  Finally, are you familiar with the idea that the "Golden Ratio" is supposed to evoke our deepest aesthetic sensitivity for what we label as beautiful?

If so, thanks for playing along. As for me, I have a lot of work cut out for me now because the concepts I referred to above embody some half-truths. I have seen them applied multiple times in the area of Christian apologetics and, thus, should be corrected, lest we embarrass ourselves (not that I think that what I write here will have a widespread effect). It's going to take me a little time to bring up all of the necessary background so that I can make my point without simply throwing one dogmatic assertion against another one, as Hegel might say. Needless to say, doing so means distributing my discussion over a number of entries, particularly since I want to commit myself to keeping any single entry from becoming too large. (I know, "promises, promises ... ")

Livio Golden RatioThe book to which I made reference in the last entry is Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (New York: Broadway Books, 2002). I will return to it for a lot of detailed information in a while.

Before going on, let me clarify that I'm going to lapse into anthropomorphisms from time to time, talking about numbers as though they were personal causal agents. Please understand that doing so is pure whimsy on my part, and that I do not ascribe any powers to numbers beyond their mathematical contexts. For example, I don't do Gematria or think of numbers as lucky or unlucky. The latter concern, it seems to me, becomes impossible because different cultures have differing numbering systems and opinions about numbers. A number that is considered auspicious in one culture, may be thought of as potentially harmful in another one, provided that it is even expressible there, and there can be no standard to measure the truth of something that's supposed to be beyond reason and logic. 

Over the last dozen years or so, I have become increasingly fascinated by the beauty of numbers, which I believe to be God's creations. Each one has its own properties; no two of them are alike. --- Okay, I know there's not a whole lot of profundity in that observation as it stands. Come to think of it, there isn't any. --- However, I'm thinking beyond mere tautologies, e.g., not just that 1 is not 2, etc.  I'm also including the special attributes that come with each number, which you may not pick up on until you have spent some time in their company. For example, look at all the things that our friend Zero can do (and, for that matter, what she's not suited for).  

Smiling ZeroWe will definitely miss out on Zero's charm if we merely think of her as "nothing." It is said that some of the early Greeks would not recognize a concept such as Zero because doing so would imply treating nothing as something.

[E.g., Charles Seife, Zero: The Biography of a Dangerous Idea (New York: Penguin, 2000) BTW, this book has some good points, but is rife with many common false generalizations concerning Greek philosophy, the Middle Ages, Aristotle's standing in the Middle Ages, the Church, and whatever else continues to play to pseudo-intellectual audiences].

Alas, the grammars of IndoEuropean languages with which I am familiar, make it unavoidable to speak of "nothing" as though it were an object. Still, even though it doesn't make sense to say that "non-being exists" (i.e. that it has being), that fact doesn't imply that it has no reality. It may not exist in the same way as pumpkins or spiders do, but it can certainly be real. If you open your refrigerator in order to get some food, and it's empty, you're confronted by something real, even though what is real is the absence (or privation) of food. So, it is certainly appropriate that in the process of counting we usually record the absence of something by labeling it with a "0." But that's only one attribute of Zero.

Zero is a place holder in our system of numbers. I've mentioned different systems before, and I'll just refer you to a collection of some of those entries. The point at hand is that in our Indian/Arabic numbering system there's a big difference between the numbers 54 and 504. In the case of the second number, the presence of Zero between the ciphers 5 and 4 has added 450 units to the number. She's telling us that the column of 10s is not represented in the numeral, which is a far cry from thinking of her as just "nothing."  

Moving on, we call on Zero to express "additive identity."

12 + 0 = 12.

We're adding Zero to 12 and still have the same old 12.

In fact, Zero does this for every number. It assures us that, if added to any number, the number remains identical to what it was before the addition. --- "Big deal," you say. "That's just playing word games." --- "Not quite," I must reply. Think of how this property works out in, say, an algebraic equation.

Let's say that you're multiplying two polynomials, e.g., (x+10) and (x-10).

Their product is  x2+10x-10x-100.

Furthermore, we know that 10x -10x=0.

Now we're definitely not fussing around with "nothing" when we recognize that the presence of 0 will not affect the identity of the rest of the formula, allowing us to eliminate Zero safely, viz. without having to maintain either "10x-10x" or "0" in our thoughts or on our papers. If you want to get to know the real Zero, in the long run it will be better to think of her as representing "additive identity" rather than as "nothing."

Queen ZeroIn the kingdom of multiplication, Queen Zero rules! Wherever she appears in a product, she takes total control, regardless of how many other factors are present or how large they are. The number of hydrogen molecules in the universe is equal to the number of moons circling the earth, once they have both been multiplied by Her Majesty Queen Zero.

On the other hand, in the adjoining land of division, Zero plays the part of an obstreperous brat. Either we give her everything (which we are not about to do), or she won't play at all. One could make use of the sleight of hand that mathematicians call "approaching the limit" to argue that any number divided by zero yields infinity, and I must state for the record that a feat along such lines is fundamental to the theory behind calculus. However, in the usual arithmetical sense, dividing by zero is not just something that you should not do, but something that you cannot do. I mean, can one even conceive of what dividing by zero means?

One more unique aspect of Zero's personality is her wanderlust. She comes and goes within the community of numbers depending on the circumstances. If you draw a line of integers moving from say -5 to 5, you will find her situated comfortably between -1 and 1. But that's only for "counting numbers," viz. cardinal numbers. For example, if we want to count the number of moose in my backyard, we'll start with 0, and--under normal circumstances--stop there as well. If we were to drive to Yellowstone Park and count how many moose we have seen while there, we'll have to start with 0, but hopefully move up from there along the ladder of integers. However, if we leave the cardinal numbers and step over into the realm of the ordinal numbers, we find that Zero is no longer with us. We can talk about the 1st moose we saw, perhaps followed by the 2nd, 3rd, 4th, etc., but it doesn't make any sense to say that before we started out we saw the 0th moose.  (The same thing applies to bears, elk, geysers, tourists, and forest rangers.)

What's more, Zero is also absent in slightly hidden applications of the concept of ordinal numbers. Take, for example, our division of history into two segments, usually designated as BC (Before Christ) and AD (Anno Domini -- the year of our Lord) or some supposedly neutral labels, such as BCE ("Before Common Era") and CE ("Common Era"). We need not occupy ourselves with the question of whether the calculations that went into setting up this system were accurate. The point is that, whatever year was thought of as the year of Christ's birth became the year 1 AD (or AD 1 for purists). It's predecessor was "Year 1 BC." There was no "Year 0," and, contrary to Seife who sees this fact as a part of the intellectual impoverishment of the Church, is how it needs to be. 

I'll stop here with my observations on Zero. We could go on in the same way, sliding up to 1, and we would find out some really entertaining properties for that number. The same goes for 2, as well as for 3, 4, 5, and really "everyone" else present at our little party for numbers. But we'll postpone such ruminations for now and try to get back to 1.61803... Here is a number that's not only unique in an obvious sense, but that's downright eccentric.



Yesterday was Halloween (as well as Reformation Day). Once again my puppets got to sit on the front porch as decoration for the festivities, and the children really liked them. For the most part, they're "monsters" in the generic sense, but friendly-looking. I am, of course, referring to the puppets, not the children, though they were also friendly-looking, even in their disguises, and they were exceptionally polite. It's always fun to watch the older ones, say, seven years old, look out for their younger siblings, who might be four or five. All but the oldest kids (like high school age) were accompanied by adults, some of whom stayed in the background, though some also came to the door. Yes, some accompanying parents even had their own collecting bags, and did not mind accepting a Snickers or Kit-Kat. A lot of the trick-or-treaters came in motor vehicles, obviously car-pooling from out-of-town.

The official time for trick-or-treating here in Alexandria was 6 pm to 9 pm. By 5:55, the first installment of about twelve kids of various ages had already found their way to our front porch. Most of them were girls in princess costumes, but I couldn't really keep track because I needed to focus on making sure that everyone got their share of goodies without my dropping either of the two big bowls in which they resided. Two lollipops and one candy bar was this year's formula. Among the various visitors there was one adult couple who was taking their sleeping baby around; the infant couldn't have been a year old yet. They looked like they were just having a lot of fun. Then there was a young man, in his early twenties I would guess, also carrying a sleeping child who appeared to me to be just a few months older than the previous one. Somehow I got the feeling that there was a back story that involved a little bit (or maybe a lot) of hurt, and June picked up the same vibrations.  All in all, we had sixty or more callers, measuring by the amount of candy that was left by 8:30 when things had become quiet. I turned off the porchlight and brought the puppets back in, thankful for the parents that cared for the safety of their children as well as for the apparent fact that good old Smalltown USA is still a place where kids can go trick-or-treating in safety. And I hope that our house will continue to be known as child-friendly in future years.


On Thursday (10/29), I finally had my first physical therapy session after my mini-stroke, which had occurred on 9/8. It's going to be challenging, but I'm sure it will be helpful. The goal is to restore some strength to my left side and my sense of balance. Neither one has been right for many years now, but the incident aggravated the problems, and I hope we can maybe get back to pre-TIA conditions.

Oh, yeah, I should probably mention that June got attacked by a fair-sized alligator a little while ago, as you can see in the picture, but she's doing fine.

June with Alligator

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Wednesday, October 28th 2015


Back to Normal? Stuff about Numbers


Life goes on.

Just for fun (at least for now), let's try something that I learned from a book I read a few years ago. I'm not going to give you its title or author today for two reasons: 1) doing so could spoil the fun for those who, along with me, know what's coming next; and 2) it could bring up some immediate misconceptions that I'm going to try to counter in the long run. Let's just do a little happy "number magic." I'm going to ask you to do a bit of arithmetic, and then I shall reveal your solution to you.​

1. Unless you have exceptional ability in arithmetic, I suggest that you use a calculator of some sort. Pencil and paper are fine, but require a bit more effort. Spreadsheets are ideal. As long as you insert the right formulas, they will do the calculations for you. Then you can also play around with different numbers without redoing any of the work. For all that I know, it may be possible to use an abacus for this procedure, but I, for one, would first have to learn how to manipulate one.

2. Pick a number, any number!

3. Pick another number, any number!

I must warn you that it will probably be best if you start with some fairly low numbers, perhaps single digit integers, because we're going to do a lot of adding. I'm saying this only to keep things manageable. This "game" works to some extent with all rational numbers, including fractions and negatives.

I'm going to provide an example, and I'm picking the numbers 9 and 12.

[Spreadsheet: a1: "9"; a2: "12" Don't put in the quotations marks!].

4. Add your two numbers.

E.g., the sum of my two numbers is 21.

[Spreadsheet: a3: "=a1+a2"  Don't forget to include the "equals" sign "="!]

5. You now have three numbers. Of those, add the last two.

For my example, I have 9, 12, and 21. The sum of the last two is, of course, 33.

[Spreadsheet a4: "=a2+a3" If you know your way around spreadsheets, you probably know how to pull down a formula by using the little handle on the cell.  Cover all of  the applicable cells; the program adjusts the cell references automatically. If that doesn't work for you, you can still write the appropriate formulas for each row and enjoy the benefit of flexibility once your column is all set up so that you can try out different numbers.]

6. Go on with the process of adding the last two numbers step by step.

E.g., I now add 21 and 33, to get 54. Then, in the next step, I will add 54 + 33 = 87, and so forth.

7. Do so for twenty numbers. I warned you: the size of the numbers will grow pretty quickly.

In my example, the nineteenth and twentieth numbers are 10,713 and 17,334

8. Now divide the 20th number by the 19th.

E.g., I need to divide 17,334 by 10,713.

[Spreadsheet: a21: "=a20/a19"]

9. Click on the link below to see your result. I'm going out to 5 decimal places; after that, the sixth decimal may differ in this quick example, perhaps due to your choice of numbers or maybe because your calculator rounds up or down. It's going to be either 3 or 4. (Hang on to that thought for later.)

I promise you that all of this is going to lead to some worthwhile thoughts with philosophical--and even theological-- impact.


On a personal note, even though my stroke was of the so-called mini variety, it did leave some residual effects, which have been getting worse over what is now close to two months from the initial event. I was supposed to have received physical therapy as soon as I left the hospital, but some incredible bureaucratic snafus have kept it from coming to pass. Needless to say, that state of affairs has exacerbated my ever-present liability for depression, which, in turn, hasn't helped my recovery either. Finally, I'm slated to have my first real session of PT, viz. not just another evaluation, tomorrow. I'm happy about that, but I also ask for your prayers that some things that could have been undone more easily six weeks ago, will still be correctable.

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Tuesday, September 22nd 2015


"Well-umm ... " and Nolo Contendere

  • STATE OF EXISTENCE: homebound

Pretty soon my blog should have more conceptual content again, as well as pictures, but at the moment it must serve merely as vehicle to report on my medical adventures to anyone who might be interested. Actually, the digressions in this post are longer than the report. I hope that the style will be entertaining enough for my patient readers to get through it, even if the content is fairly weak. A note to any new readers: It is very seldom that my tongue is not in my cheek when I report on myself.


“He hasn’t been driving, has he?” Dr. G. turned his head away from me, perched on the examination table, towards June, seated on a little wooden chair in the corner. One or both of us replied, “Well-umm …”

Let me inject here a quick observation on the phrase “Well-umm …” Despite its innocuous appearance, it plays a significant role in American conversational English. If I remember correctly, it was the late actor McLean Stevenson who once described his early career as consisting of “Well-umm …” roles, viz. his characters would be surprised by some event or question, and he would barely have time enough to say "Well-umm …” before the more important characters would take over the scene.

But its meaning goes beyond covering up a moment of hesitancy. It is roughly synonymous with the more formal nolo contendere, the plea made famous by the late Spiro T. Agnew (1918-1996), known as "Ted Agnew" the moderate Republican governor of Maryland (‘67-‘69) who reinvented reinvent his image as "Spiro Agnew," the conservative Republican U.S. vice president (’69-’73). His Maryland campaign jingle, “My Kind of Man, Ted Agnew is,” will probably always stick in my mind, as well as some of his alliterated put-downs of political opponents, e.g., “the nattering nabobs of negativism.” (The phrase was actually concocted by William Safire. Click here for its common revisionist description and assessment, and here for an analysis of its actual occurrence and significance.)

Hmm. It looks like I’m at least sufficiently back in form to engage in my usual practice of “stacking.” So, I need to remember now to pop back to the topic of the moment, the phrase “Well-umm … “ I would suggest that the person using the phrase is making what J.L. Austin (1911-1960) called a “performative utterance." Instead of declaring anything, the speaker is mumbling an inanity that really serves as an admission of guilt. It is closely related to, “Well, you know …” and “Yeah, ah, yeah, about that …”

Not that we realized before that moment today that there was any “guilt” involved, having faithfully followed and defended the doctor’s other orders. It hadn’t occurred to me that in my present stage of recovery I should not be piloting a ton of metal on wheels through traffic. Duh! We had not made the longer trip to Tennessee, but I had driven us on several short shopping excursions. Needless to say, the driving prohibition does make sense, but it also makes life a little more complicated for us, particularly since June is not very comfortable behind the wheel any more. Still, Dr. G. instructed us that I am not to drive for at least another two more weeks and should consider myself “homebound.”

I must say that Dr. G has surprised me a little by how seriously he really is taking what happened to me, but, as he explained, he has seen too many people have a mini-stroke, wave it off as soon as they think they feel better, and wind up with a major stroke not too long thereafter. So, in addition to taking all my meds, which I am, and staying away from tension or pressure, which I’m trying to do as much as it is under my control, I’m also not supposed to drive for at least the next two weeks. This directive is being reinforced inasmuch as Dr. G’s office will schedule some physical therapy for me as a “homebound” patient, which means that the therapist will come to our house, something that really makes me feel just a little older than my usual self-perception. Finally, our good doctor commended June on doing her job as my “guard,” a quite thankless role at times over these last couple of weeks.


Win on front porch

Thanks to those who gave a few moments of their time for my front-porch StreetJelly attempt last Sunday. Alas! I had no sooner started the set, when the gentleman across the street revved up his lawn mower. And when I thought he was done, he cranked up his gas-powered weed whacker, which exceeded the lawn mower in decibels. Time to cue “Pleasant Valley Sunday.” (This video is by Carole King who co-wrote the song; I couldn’t find a listenable Monkees’ version.) Anyway, sorry about the assault on your ears if you tuned in on Sunday to my impromptu set in the open air of Smalltown, USA. I’m planning on doing my usual StreetJelly show this coming Thursday at 9pm Eastern. Since it’s the fourth Thursday of the month, it’ll be an all-gospel program.


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